Rational and Irrational numbers
Section: Number and Calculation | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
Rational Numbers
Every number you meet in school falls into one of two families: rational or irrational. Telling them apart comes down to a single test - can the number be written exactly as a fraction of two integers?
- A rational number is any number that can be written as a fraction p/q, where p and q are integers and q ≠ 0.
- Every integer is rational, since any integer n can be written as n/1.
- A decimal is rational if it either terminates (ends after a finite number of digits, e.g. 0.5, 0.75, 3.125) or recurs (one or more digits repeat forever, e.g. 0.333... = 0.3̇, 0.454545... = 0.4̇5̇).
| Number | As a Fraction | Why it's Rational |
|---|---|---|
| 5 | 5/1 | All integers are rational |
| 0.5 | 1/2 | Terminating decimal |
| 0.333... | 1/3 | Recurring decimal |
| -2.75 | -11/4 | Can be written as a fraction |
| 0 | 0/1 | Zero is rational |
Worked Example: Classifying Numbers as Rational
-
Question: Which of these numbers are rational: 8, −3/5, 0.36, 0.101001000100001... (the digit pattern never repeats)?
- Step 1: 8 = 8/1, so it is rational.
- Step 2: −3/5 is already a fraction of two integers, so it is rational.
- Step 3: 0.36 terminates after 2 digits, so it is rational.
- Step 4: 0.101001000100001... never settles into a repeating block, so it is not rational.
- Answer: 8, −3/5 and 0.36 are rational; 0.101001000100001... is irrational.
Irrational Numbers
An irrational number refuses to settle into a fraction or a repeating decimal pattern, no matter how far you calculate it. Because of this, irrational numbers can only ever be approximated - written exactly, they go on forever with no pattern.
- An irrational number cannot be written as a fraction p/q - its decimal representation is non-terminating and non-recurring.
- Not all square roots are irrational: √4 = 2 and √9 = 3 are both rational. Only square roots of non-perfect squares are irrational.
| Number | Symbol | Approximate Value |
|---|---|---|
| Pi | π | 3.14159265358979... |
| Square root of 2 | √2 | 1.41421356237... |
| Square root of 3 | √3 | 1.73205080756... |
| Euler's number | e | 2.71828182845... |
| Golden ratio | φ | 1.61803398874... |
Irrational numbers can be located approximately on a number line, but never landed on exactly
Worked Example: Deciding if a Square Root is Rational
-
Question: Is √50 rational or irrational?
- Step 1: Check the perfect squares either side of 50: 7² = 49 and 8² = 64.
- Step 2: 50 is not equal to either, so 50 is not a perfect square.
- Step 3: The square root of a non-perfect square is always irrational.
- Answer: √50 is irrational (√50 ≈ 7.07)
Estimating Square and Cube Roots (Surds)
- A surd is a root - such as √2 or ³√5 - left in root form because its exact value is irrational and cannot be written as a terminating or recurring decimal.
- To estimate a square root, find the two consecutive perfect squares it lies between, then judge which one it is closer to.
- To estimate a cube root, use the same idea with consecutive perfect cubes instead.
Worked Example: Estimating a Square Root
-
Question: Estimate √40 to 1 decimal place.
- Step 1: Find the surrounding perfect squares: 6² = 36 and 7² = 49, so √40 lies between 6 and 7.
- Step 2: 40 is much closer to 36 than to 49, so √40 is closer to 6 than to 7.
- Step 3: A reasonable estimate is 6.3, since 6.3² = 39.69, very close to 40.
- Answer: √40 ≈ 6.3
Worked Example: Estimating a Cube Root
-
Question: Between which two integers does ³√100 lie?
- Step 1: Find the surrounding perfect cubes: 4³ = 64 and 5³ = 125.
- Step 2: Since 64 < 100 < 125, ³√100 must lie between 4 and 5.
- Answer: ³√100 is between 4 and 5
Multiplying and Dividing Surds
- Two surds can be combined into a single root: √a × √b = √(ab) and √a ÷ √b = √(a ÷ b).
- If the number left under the root is a perfect square, the surds combine to give a whole number - two "irrational-looking" surds can multiply or divide to give a rational answer.
- The same idea extends to cube roots: ³√a × ³√b = ³√(ab). For example, ³√2 × ³√4 = ³√8 = 2.
Worked Example: Multiplying Surds
-
Question: Calculate √5 × √45.
- Step 1: Combine the two roots into one: √5 × √45 = √(5 × 45)
- Step 2: Multiply the numbers underneath: 5 × 45 = 225
- Step 3: 225 is a perfect square, since 15² = 225
- Answer: √5 × √45 = 15
Worked Example: Dividing Surds
-
Question: Calculate √98 ÷ √2.
- Step 1: Combine the two roots into one: √98 ÷ √2 = √(98 ÷ 2)
- Step 2: Divide the numbers underneath: 98 ÷ 2 = 49
- Step 3: 49 is a perfect square, since 7² = 49
- Answer: √98 ÷ √2 = 7
Key Differences Between Rational and Irrational Numbers
Side by side, the two families of numbers behave in opposite ways on every test that matters - whether they fit a fraction, what their decimal looks like, and whether they can be pinned to an exact spot on a number line.
| Feature | Rational Numbers | Irrational Numbers |
|---|---|---|
| Fraction form | Can be written as p/q | Cannot be written as p/q |
| Decimal form | Terminating or recurring | Non-terminating and non-recurring |
| Examples | 1/2, 0.75, 3, -5, 0.3̇ | π, √2, √3, e |
| On number line | Can be located exactly | Can be approximated |
The Real Number System
Every rational and irrational number you have met so far is a real number. Zooming into the rational side reveals a further nested hierarchy, each family sitting entirely inside the one before it:
Each family of numbers is completely contained inside the next one out - every natural number is a whole number, every whole number is an integer, and so on
Interactive revision notes, videos and practice questions load below.